ANNALES
UNIVEBSITATIS MARIAE CUBIE-SKŁODO W8KA LUBLIN - POLONIA
VOL. XXII/XXIII/XXIV, 16 SECTIO A 1968/1969/1970
Instytut Matematyczny Polskiej Akademii Nauk, Pracownia w Łodzi
JULIAN ŁAWRYNOWICZ and WŁODZIMIERZ WALISZEWSKI
Conformality and Pseudo-riemannian Manifolds Konforemność i rozmaitości pseudoriemannowskio Конформность ii псевдорнмановые многообразия
Introduction
Conformal mappings of Riemannian manifolds were investigated by several authors in the local and global formulation as well (cf. e.g. [3], [8], and [10]), and some results were also obtained in the case of pseudo- riemannian manifolds, however, in the local formulation only (cf. e.g.
[3], [9], and [7]). Quasiconformal mappings of Riemannian manifolds were introduced and investigated in [13].
In the present paper we are concerned with conformal mappings of pseudo-riemannian manifolds in the global formulation.
We begin our study with preliminaries. We introduce first some notation and terminology, in particular the notion of an essentially pseudo-riemannian manifold, develop measurability and integration (Theorems 1 and 2), introduce the notion of an angle, and define its inner measure. We deal then with curves, especially we distinguish some kinds of curves: space-like, time-like, regular, and rectifiable, define the length of a regular curve, introduce some kinds of mappings: type-preser
ving and type-reversing, and give a basic theorem on these mappings (Theorem 3). Next we introduce the notion of the p-modulus of a family of regular curves and study basic properties of these moduli (Theorems 4-8).
In the second part of the paper (Section 6) we are concerned with conformal mappings of essentially pseudo-riemannian manifolds. We introduce the notion of conformality that, roughly speaking, means that the isotropic cone is preserved at each point of the manifold in question.
We give then a necessary and sufficient condition for conformality in terms of quadratic forms determined by the metrics of the manifolds
Annales >
114 Julian Ławrynowicz, Włodzimierz Waliszewski
in question (Theorem 9). Now we give a characterization of conformal mappings in terms of angles and their inner measure (Theorems 10 and 11), and, finally, in terms of families of regular curves and their moduli (The
orems 12 and 13).
In the last section we define regular quasiconformal mappings and conclude the paper with the result that in the case of essentially pseudo- riemannian manifolds there is no analogue of regular quasiconformal mappings other than conformal. Here we mention that the problem of the existence of some irregular quasiconformal mappings remains open.
We also pose some other natural problems, some of them being planned to be discussed in a subsequent paper.
The authors are deeply indebted to Dr Kalevi Suominen for helpful discussions and suggestions during their research in the subject in question.
The detailed version of this paper will appear in the Mathematica Scandinavica 28 (1971).
1. Notation and terminology
Throughout this paper the set of all points (resp. vectors) of a manifold (resp. vector space) X is denoted by suppA. If f is a mapping from a set (resp. manifold or a vector space) X into a set (resp. manifold or vector space) T, we write /: X -> T, and denote the image of any subset E of X (resp. suppA) by f[E]. If, in particular, f is a homeomorphism, -> means “onto”, i.e. T = /[X] (resp. suppA =/[suppA]).
The «.-dimensional Euclidean space is denoted by B", and its subspace that consists of points with the last component positive — by B" . In the case where n = 1 we drop out the index n.
Under a pseudo-riemannian manifold we mean a C°°-differentiable paracompact connected manifold endowed with a pseudo-riemannian metric, i.e. a symmetric C°° tensor field of type (0,2) which is nondege
nerate and has the same index at each point. Let g be the metric in question.
Denote by n and p its dimension and index, respectively. Clearly, there is no loss of generality if we assume that p < j«., i.e. if we replace, if convenient, g by —g. We say that a pseudo-riemannian manifold is essentially pseudo-riemannian if l<p<|«. For the definition and pro
perties of C00-differentiable manifolds as well as tensor fields we refer to [1].
Given a pseudo-riemannian manifold Jf and an a?e supp Jf, T^JZ denotes the tangent space to Jf at x, while
Z® JZ = {i)e suppT^JZ: g(v, v) — 0}, Z/Jf = {vc svppTxM: g(v, v) > 0}, I~M = {r e suppZ'r JZ: g(v, v) < 0}.
Conformality and pseudo-riemannian manifolds 115 In other words 7®M is the collection of vectors of all isotropy sub
spaces of TrM, while ZJAf and Z~Jf are the collections of vectors of all positive and negative definite subspaces of TXM, respectively. Further, TM denotes the tangent bundle of M. Finally, if N is another pseudo- riemannian manifold and /: M -* N a diffeomorphism, then Df: TM -> TN denotes the derivative of /.
2. Measurability and integration
Suppose that A and A are C°°-differentiable paracompact connected manifolds, while M and N are pseudo-riemannian manifolds with metrics g and g', respectively. Under a Borel measure on X we mean a measure which is defined on the collection of Borel subsets of suppA. A mapping /: X -> Y is said to be a Borel f unction if the preimage [72] of each
open set E <= supp A is a Borel set.
As in [13], p. 8, we say that a set E c suppA' is a null set if for each coordinate neighbourhood U c suppA and each coordinate C°°-mapping //: U -> supp 77" the set y[Er\U] has Lebesgue measure zero. A condition is said to hold for almost every are suppA, or almost everywhere on A, if it holds everywhere except perhaps for a null set. In our considerations as derivatives of functions are differentiable almost everywhere we shall meet functions which are not defined on a Borel null set. If such a function is Borel on its set of definition, then its extension by a constant value will also be Borel. We will carry out always such an extension by the value 0. Hence we may regard all functions as defined everywhere.
Theorem 1. Suppose that f: M -> N is continuous and differentiable almost everywhere. For any xe supp M consider arbitrary coordinate C°°- -mappings y — (/?) on M at x and v = (vr) on N at f(x) whose dimensions are equal to the dimensions of the corresponding manifolds. Then
(1) the quantities
||Z)/||(a;) = s\xp\g'[Df(x)(v), Df (x) (v))\il2, xe supp M,
where the supremum is taken over all veTxM such that |</(®, ®)| 1, and
. , I det oL or of (a:) 1/2
(detDf)(x) = det(Jo fey ),<o/z(ar)! detg » xe 8UPPM, where |t denotes partial differentiation with respect to y*, do not depend on the choice of y and v,
(ii) the functions \\Df\\: M -> R and detZ)/: M -> R are Borel.
We introduce then the notion of jacobian. If f: M -> N is a CMiffe- omorphism, then, by Theorem 1, detZ>/ is a real-valued Borel function
Julian Ławrynowicz, Włodzimierz Waliszewski 116
of a?e supp M, i.e. detJ>/: M -> R. Moreover, as it is easily seen, it is con
tinuous. The function
Jf = |detD/|
is called the jacobian of f.
The following theorem enables us to define the Lebesgue measure on a pseudo-riemannian manifold.
Theorem 2. With each M we can associate a unique Borel measure r(Jf) so that the following conditions are satisfied:
(i) if N is an open pseudo-riemannian submanifold of M, then
= r(N)(E) for all Borel sets E c. suppN, (ii) if f: M-^N is a (f-diffeomorphism, then r(N)(f[E]) = f J,dr (HI)
E
for all Borel sets E c supp Jf,
(iii) if M — Bm or JS”, m =1,2,..., then r(M) is the Lebesgue measure.
Now we define the Lebesgue measure on a pseudo-riemannian manifold Jf as the measure r(Jf) determined in Theorem 2.
We conclude this section by a corollary.
Corollary 1. If f: JI-> JV is a C^diffeomorphism, then a Borel function q: N -> It is r(N)-integrable if and only if (oof) J, is r(Jf)-
integrable and
f edr(W) = f (eof)Jfdr(JH).
w M
The proofs are analogous to that given in [13], p. 9-12, in the case of Riemannian manifolds.
3. Angles and tlieir inner measure
Let Jf be an essentially pseudo-riemannian manifold with metric g.
For a real number a, a yt 0, let
1“ Jf = {«« supp Ta, Jf: g(v, v) = a}.
We say that a set E forms an ordinary angle arg(a>, E) at a point a; of Jf if E is a Borel subset of some I"M, a 0. We say that a set E forms a topological angle arg(ic, E) at a point x of Jf if E is a Borel subset of either I+Jf or Ix^.
Let xe supp Jf. Given a set E that forms a topological angle at x, let 1XE = {bv: ve E,0 <b < l/\g(v, u)!172}.
Conformality and pseudo-riemannian manifolds 117 It is easily seen that IXE is Lebesgue-measurable on TXAI. According to Section 2, we denote its Lebesgue measure by t(TxAI)(IxE) and the volume element by dr(TxAI). We then define the inner measure A(x, E) of arg(®, E) by
A(x,E) =t(TxM)(IxE).
4. Curves and arc length
Let AI be an essentially pseudo-riemannian manifold with metric g.
Under a curve on AI we understand a continuous mapping c from a closed interval [a; 6], b, to AI. If c is differentiable, we identify the deriva
tive Dc(t), te [a; 6], with a tangent vector to Af at c(t). This determines a curve Dc in the tangent bundle TAI.
A curve c is called space-like (resp. time-like) if it is absolutely con
tinuous and Dc(t) is a vector of a positive (resp. negative) definite subspace of at every point of differentiability. If c is either space-like orl time-like, it is called regular.
The length of a regular curve c is defined by 1(c) = f |ff(Dc(0,Dc(<)|l/2dL
If 1(c) is finite, c is said to be rectifiable. Now let g: AI -> It be a Bore function, c0 — the parametrization of c by arc length, and ds — the arc length element. The integral of q along c is defined by
H«)
J ds = J goc0ds,
c o
provided that the latter integral exists. Otherwise the integral of q along c is undefined.
Finally, suppose that N is an essentially pseudo-riemannian manifold and f: AI -r N a C1-diffeomorphism. Then f is said to be type-preserving (resp. type-reversing) if it transforms space-like curves onto space-like (resp. space-like) curves. Here we confine ourselves to one theorem needed later on:
Theorem 3. Suppose that f: AI -^ N is either type-preserving or type- -reversing, c: is rectifiable, while q: N -*■ R is Borel and non
negative. Then f(c) is rectifiable and
f eds^ j(QOf)\\Df\\dS.
ftp) o
The proof is analogous to that given in [13], p. 14, in the case of Bie- mannian manifolds.
118 Julian Ławrynowicz, Włodzimierz Waliszewski
5. Moduli
Here we give an analogue of the p-moduli discussed in [13], p. 15-20.
Our composition and proofs, however, follow rather [4] or [11]. Through
out the whole section Jf is an essentially pseudo-riemannian manifold, while C, Co, Cx, C2, ... are families of regular curves on M.
Denote by admO the class of all nonnegative Borel functions q on Jf which satisfy
J g ds 1 c
for all rectifiable ce C. Here we do not assume that the integrals in question are finite. If ge admC, q is said to be an admissible metric for C.
For each positive number p we define the p-modulus modJ(C of C by modpC = inf ) gpdr,
31
where the infimum is taken over all ge admC. If admC is empty, we put modpC — oo. The quantity l/modpC is called the p-extremal length of G.
If in admO there is a metric g0 such that modpC = J Q%dr,
m
then p0 is called p-extremal. It has the following important property:
Theorem 4. (uniqueness of an extremal metric). 2/, for some positive integer p, modpC is finite and g0, q* are p-extremal, then g* — q0 almost everywhere on M.
Now we formulate other basic properties of p-moduli. Thereafter p is a positive number and 27, u denote summation over all positive integers ft.
Theorem 5 (monotoneity of moduli). If Ck c C2 or, more generally, each cx of C1 contains a c2 of C2, then
modpCj < modpCjj.
Theorem 6 (the principle of composition for extremal lengths).
Suppose that Ck,k =1,2,..., consist of curves lying in disjoint Borel subsets Ek of supp M, respectively, and that any c of C contains some curve of Ck for each k. Then
i i
l/modj'~1 Ck < 1/modJ'-1 C, p > 1,
1 /modp 6\. < 1 /modp C,p > 2.
GonformaUty and pseudo-riemannian manifolds 119 Theorem 7 (subadditivity of moduli). If C = {J Cki then
modpC < V*modpCfc.
Theorem 8 (superadditivity of moduli), (i) Suppose that (J C,.c C and that all Gk consist of curves lying in disjoint Borel subsets Ek of supp If, respectively. Then
(1) 5 modpCfc < modpC.
(ii) Estimate (1) remains valid if the condition J Ck a C is replaced by the requirement for each ck of Ck, k = 1, 2, ..., to contain some curve of C.
Theorems 4-8 are valid also in the case where M is pseudo-riemannian but not essentially. If the index p of Jf equals 1, i.e. in the riemannian case, C, 6’0, C\, C\, ... denote just families of curves on Jf (cf. [13], p. 15-20). If %n < p < n, we can establish the same results as those given above on replacing the metric g of Jf by — g.
6. Conformality
Thereafter we always assume that M and N are essentially pseudo- -riemannian manifolds with metrics g and g', repsectively, while/: Jf-> Jf is a C'-diffeomorphism.
A C’-diffeomorphism /: JI -> N is said to be conformal if
(2) Df (x) [/“M] = N,xe supp Jf,
in the case where the index of g is less than %n, while (3) D/(.t)[7+Jf] = IfaN, xe supp Jf,
in the case where the index of g equals in, n being the dimension of g.
We begin with a theorem that gives a necessary and sufficient con
dition for conformality. This condition agrees with the usual definition applied in the case of Riemannian manifolds (cf. [8], p. 106, [10], vol. I, p. 309, and [13], p. 16) as well as in the local formulation in the case of pseudo-riemannian manifolds (cf. [3], p. 89, and [9], p. 5).
Theorem9. A C'-diffeomorphism ft M -> N is conformal if and only if g'(Ef(x)(v), I>f(x)(v)) = a (as) 0 O’, v),
a(x) > 0, Xe supp Jf, ve supp TV,. M, where a does not depend on v.
Theorem 9 implies:
Corollary 2. Conditions (3) and
Df(x)[I~ Jf ] = If N,xe supp M,
120 Julian Ławrynowicz, Włodzimierz Waliszewski
are both necessary and sufficient for a Cl-diffeomorphism f: M -> N to be conformal.
Next we give a characterization of conformal mappings in terms of the inner measure of angles.
Theorem 10. If f: M -> N is conformal and E forms a topological angle at xe supp Jf, then
(i) Df(x)[E] forms a topological angle at f(x) and A(f(x), Df(x)[E]) — A(x, E),
(ii) the relation E a If M implies Df(x)[E] <= IfX)E, while EcI~M imples Ef(x)[E] c
If, in particular, E forms an ordinary angle at x, then Df(x)[E] forms an ordinary angle at f(x).
Theorem 11. Suppose that f: M->N is a Cl-diffeomorphism and that if E forms an ordinary angle at xe supp M, then
(i) Df(x)[E] forms a topological angle at f(x),
(ii) the relation E <= If M implies Df(x)[E] c If^N, while E c. IfM implies Ef(x)[E] c If^N.
Then f is conformal.
Finally we give a characterization of conformal mappings in terms of moduli.
Theorem 12. If f is conformal, then it is type-preserving. Furthermore, if C is a family of regular curves on M, then
(4) mod„/(C) = modnC.
V, in particular,
0 < k < ||P/(®)|| < K < oo, xe supp M, then
K*~pmodpC < modp/fC) < kn~pmodpC for p^ n and
kn~pmodpC < modp/(C) < K" “pmod:PC' for p ^.n.
Theorem 13. If f: M -> N is type-preserving, then it is conformal.
7. Conclusions
Suppose that/: № -> N is type-preserving and that there is a constant Q,1^Q < oo, such that
(5) (l/QJmodnO < mod„/(C) < QmodnC
for some family C of regular curves. Then, by Theorem 13, / is conformal and consequently, by Theorem 12, we get (4). Hence we conclude that in the case of essentially pseudo-riemannian manifolds there is no analogue
Conformality and pseudo-riemannian manifolds 121 of regular quasiconformal mappings other than conformal (cf. [12], p. 18, 179, and 222 (Theorems 3.2 and 4.2), for the plane case; [14], p. 18-19, for the euclidean case; and [13], p. 24-25, for the riemannian case).
Nevertheless, it is quite possible that if we properly weaken the hypotheses of Theorem 13 in the sense that we allow some less smooth mappings and assume, in addition, that f preserves the «-moduli, we will still be able to prove that f is conformal (cf. [5], p. 388-390). Then it will be natural to consider also the case where the preservation of the «.-moduli is replaced by a quasi-preservation in the sense of (5) with some fixed Q, where G ranges over the class of all families of regular curves on Jf.
Other important problems that seem to be very natural are the convergence properties of sequences of conformal mappings, in particular, the problem of finding some conditions under which the limit mapping is conformal. These questions, including the problem of obtaining some analogue of the Carathćodory convergence theorem (cf. [2] and [6]), are essential for physical applications. They were not solved even in the riemannian case.
The authors plan to discuss at least some of these problems in a sub
sequent paper.
REFERENCES
[1] Bishop, R. L., and Goldberg, S. I., Tensor analysison manifolds, New York — London 1968.
[2] Car a thćodory, C., Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten, Math. Ann. 72 (1912), p. 107-144.
[3] Eisenhart, L. P., Riemannian geometry, Princeton 1949.
[4] Fuglede, B., Extremal length andfunctional complection, Acta Math. 98 (1957) p. 171-219.
[5] Gehring, F. W., Rings and quasiconformal mappings in space, Trans. Amer.
Math. Soc. 103 (1962), p. 353-393.
[6] — The Caratheodory convergence theorem for quasiconformal mappings in space, Ann. Acad. Sei. Fenn. Ser. AI 336/11 (1963), 21 p.
[7] — and Haahti, H., The transformations which preserve the harmonic functions, ibid. 293 (1960), 12 p.
[8] Goldberg, S. I., Curvature and homology, New York — London 1962.
[9] Haahti, H., Über konforme Abbildungen eines euklidischen Raumes in eine Riemannsche Mannigfaltigkeit, Ann. Acad. Sei. Fenn. Ser. AI 287 (1960), 54 p.
[10] Kobayashi, S., and Nomizu, K., Foundations of differential geometry I-II, New York — London— Sydney 1963-1969.
[11] Krzyż, J., and Ławrynowicz, J., Metoda długości ekstremalnej i jej zastoso
wania [The method of extremal length and its applications], Materiały Konferenoji Szkoleniowej Funkcji Analitycznych w Uniejowie, Łódź 1969, p. 19-83.
[12] Lehto, O., und Virtanen, K. I., Quasikonforme Abbildungen, Berlin — Hei
delberg — New York 1965.
[13] Suominen, K., Quasiconformal maps in manifolds, Ann. Acad. Sei. Fenn. Ser.
AI 393 (1966), 39 p.
[14] Väisälä, J., On quasiconformal mappings in space, ibid. 298 (1961), 36 p.
122 Julian Ławrynowicz, Włodzimierz Waliezeweki STRESZCZENIE
Praca dotyczy odwzorowali konforemnych rozmaitości psendorie- mannowskich rozpatrywanych globalnie.
W części przygotowawczej wprowadzamy pewne oznaczenia i pojęcia, w szczególności pojęcie rozmaitości pseudoriemannowskiej, dyskutujemy zagadnienia mierzalności i całkowalności, wprowadzamy pojęcie kąta i definiujemy jego miarę wewnętrzną. Z kolei zajmujemy się krzywymi, w szczególności wyróżniamy pewne ich rodziny: przestrzenne, czasowe, regularne i prostowalne, definiujemy długość krzywej regularnej, wpro
wadzamy pewne klasy odwzorowań: zgodne i niezgodne oraz uzyskujemy podstawowe twierdzenie o tych odwzorowaniach, które daje pewną nie
równość istotną dla dalszych badań. Następnie wprowadzamy pojęcie modułu rzędu p rodziny krzywych regularnych oraz badamy podstawowe własności tych modułów.
W drugiej części pracy zajmujemy się odwzorowaniami konforemnymi rozmaitości istotnie pseudoriemannowskich. Wprowadzone pojęcie kon- foremności oznacza, z grubsza biorąc, zachowanie stożka izotropowego w każdym punkcie rozpatrywanej rozmaitości. Z kolei uzyskujemy wa
runek konieczny i dostateczny konforemności w terminach form kwadra
towych określonych przez metryki rozpatrywanych rozmaitości. Podajemy charakteryzację odwzorowań konforemnych w terminach kątów i ich miary wewnętrznej, a w końcu, w terminach krzywych regularnych i ich modułów.
Wreszcie, definiujemy odwzorowania ąuasi-konforenme regularne i podsumowujemy pracę wynikiem orzekającym, iż w przypadku rozmai
tości istotnie pseudoriemannowskich nie ma odpowiednika odwzorowań ąuasi-konforemnych regularnych i niekonforemnych jednocześnie. Prag
niemy tu zaznaczyć, że problem istnienia stosownie określonych odwzo
rowań quasi-konforenmych nieregularnych pozostaje otwarty. W zakoń
czeniu stawiamy także pewne inne naturalne problemy.
РЕЗЮМЕ
Работа касается конформных отображений псевдорнмановых мно
гообразий, рассматриваемых в целом.
В предварительной части введены некоторые обозначения и тер
минология, в частности понятие псевдориманового многообразия, рассмотрены вопросы измеримости и интегрирования, введено понятие угла и определена его внутренняя мера. Далее рассмотрены кривые с особенным выделением нескольких семейств кривых: пространствен
ных, временных, регулярных и спрямляемых; определена длина регулярной кривой, введено несколько видов отображений: согласные
Сон/огтаШу ап<1 рвеийо-петапплап танЛ/оМн 123 и несогласные и получена основная теорема об этих отображениях, дающая некоторое неравенство существенное для дальнейших иссле
дований. Введено понятие р-модуля семейства регулярных кривых и изучаются основные свойства этих модулей.
Во второй части работы рассмотрены конформные отображения, по существу псевдоримановых многообразий. Введено понятие кон
формности, которое обозначает, приблизительно, сохранение изо
тропного конуса в каждой точке рассматриваемого многообразия.
Получено необходимое и достаточное условие для конформности в терминах квадратных форм, которые определены метриками рас
сматриваемых многообразий. Дана характеристика конформных ото
бражений в терминах углов и их внешней меры, а также в терминах регулярных кривых и их модулей.
В конце работы определены регулярные квазиконформные ото
бражения и сделан вывод, что в случае по существу псевдоримановых многообразий нет совместного аналога для квазиконформных и ре
гулярных отображений. Следует отметить, что проблема существо
вания соответственно определенных нерегулярных квазиконформных отображений остается открытой. Представлены также и другие естест
венные проблемы.
